Here are the answers going back to 1965. It makes sense that the share of those listing "big labor" has dropped over time, given the decline in the share of US workers belonging to a union. It's interesting that the share naming big business as the biggest threat is about the same now as in the Carter era of the late 1970s. Sure, there have been some spikes in viewing big business as the biggest threat, like a spike probably related to Enron and other corporate governance scandals in the early 2000s and a spike around 2009 in the aftermath of the corporate and financial bailouts. But there doesn't seem to be an overall trend upward. For big government, on the other hand, there does seem to be a long-term upward trend: that is, given the way the question is phrased, the shift away from seeing "big labor" as the leading issue has been counterbalanced by a swing toward naming "big government" instead.
What's the political breakdown here? Here's the share naming big government as the biggest threat, broken down by party. It's no surprise that Republicans are most likely to name big government as the problem and Democrats are least likely, with Independents falling between. It's also perhaps expected that when Barack Obama was elected president in 2008, the share of Democrats naming "big government" as the biggest threat showed a big drop. But historically, it looks as if a little more than half of Democrats see "big government" as the biggest threat through the 1980s and 1990s, and in the December 2015 data as well.
Of course, it's worth noting (especially in an election year) that opposition to "big government" might not be a broad philosophical opposition to all big government, but just be disappointment or opposition to the existing government--perhaps along with the evergreen belief that an alternative big government would perform better.
